Theorem aidm 10654

Description: The underlying directed multi graph of a deductive system.

Hypotheses

Ref	Expression
aidm.1	|- D = (dom` T)
aidm.2	|- C = (cod` T)
aidm.3	|- O = dom (id` T)

Assertion

Ref	Expression
aidm	|- (T e. Ded -> <.<.D, C>., ran D>. e. Dgra)

Proof of Theorem aidm

Step	Hyp	Ref	Expression
1		dedalg 10647	. . . . 5 |- (T e. Ded -> T e. Alg)
2		eqid 1478	. . . . . 6 |- dom D = dom D
3		aidm.1	. . . . . 6 |- D = (dom` T)
4		aidm.3	. . . . . 6 |- O = dom (id` T)
5		eqid 1478	. . . . . 6 |- (id` T) = (id` T)
6	2, 3, 4, 5	doma 10632	. . . . 5 |- (T e. Alg -> D:dom D-->O)
7	1, 6	syl 10	. . . 4 |- (T e. Ded -> D:dom D-->O)
8	4, 3	rdmob 10652	. . . . 5 |- (T e. Ded -> ran D = O)
9		feq3 3628	. . . . 5 |- (ran D = O -> (D:dom D-->ran D <-> D:dom D-->O))
10	8, 9	syl 10	. . . 4 |- (T e. Ded -> (D:dom D-->ran D <-> D:dom D-->O))
11	7, 10	mpbird 196	. . 3 |- (T e. Ded -> D:dom D-->ran D)
12	3	dmeqi 3318	. . . . . 6 |- dom D = dom (dom` T)
13		eqid 1478	. . . . . 6 |- (dom` T) = (dom` T)
14		eqid 1478	. . . . . 6 |- dom (id` T) = dom (id` T)
15		aidm.2	. . . . . 6 |- C = (cod` T)
16	12, 13, 14, 5, 15	coda 10633	. . . . 5 |- (T e. Alg -> C:dom D-->dom (id` T))
17	1, 16	syl 10	. . . 4 |- (T e. Ded -> C:dom D-->dom (id` T))
18	14, 3	rdmob 10652	. . . . 5 |- (T e. Ded -> ran D = dom (id` T))
19		feq3 3628	. . . . 5 |- (ran D = dom (id` T) -> (C:dom D-->ran D <-> C:dom D-->dom (id` T)))
20	18, 19	syl 10	. . . 4 |- (T e. Ded -> (C:dom D-->ran D <-> C:dom D-->dom (id` T)))
21	17, 20	mpbird 196	. . 3 |- (T e. Ded -> C:dom D-->ran D)
22	11, 21	jca 288	. 2 |- (T e. Ded -> (D:dom D-->ran D /\ C:dom D-->ran D))
23		ismgra 10613	. . 3 |- ((D e. V /\ C e. V /\ ran D e. V) -> (<.<.D, C>., ran D>. e. Dgra <-> (D:dom D-->ran D /\ C:dom D-->ran D)))
24	3	a1i 8	. . . 4 |- (T e. Ded -> D = (dom` T))
25		fvex 3738	. . . 4 |- (dom` T) e. V
26	24, 25	syl6eqel 1559	. . 3 |- (T e. Ded -> D e. V)
27	15	a1i 8	. . . 4 |- (T e. Ded -> C = (cod` T))
28		fvex 3738	. . . 4 |- (cod` T) e. V
29	27, 28	syl6eqel 1559	. . 3 |- (T e. Ded -> C e. V)
30	24	rneqd 3347	. . . 4 |- (T e. Ded -> ran D = ran (dom` T))
31	25	a1i 8	. . . . 5 |- (T e. Ded -> (dom` T) e. V)
32		rnexg 3365	. . . . 5 |- ((dom` T) e. V -> ran (dom` T) e. V)
33	31, 32	syl 10	. . . 4 |- (T e. Ded -> ran (dom` T) e. V)
34	30, 33	eqeltrd 1551	. . 3 |- (T e. Ded -> ran D e. V)
35	23, 26, 29, 34	syl3anc 860	. 2 |- (T e. Ded -> (<.<.D, C>., ran D>. e. Dgra <-> (D:dom D-->ran D /\ C:dom D-->ran D)))
36	22, 35	mpbird 196	1 |- (T e. Ded -> <.<.D, C>., ran D>. e. Dgra)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  <.cop 2415  dom cdm 3176  ran crn 3177  -->wf 3184  ` cfv 3188  Dgracmgra 10611  Algcalg 10614  domcdom_ 10615  codccod_ 10616  idcid_ 10617  Dedcded 10638

This theorem is referenced by:  aidmold 10655

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-mgra 10612  df-alg 10619  df-doma 10620  df-coda 10621  df-ida 10622  df-cmpa 10623  df-ded 10639

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